Flat Touch Sensor 0/1 Help For LED Lamp

I’m looking to design a power button similar to that of the Xbox One; a simple, flat 0/1 switch that’s activated by touch, not press. I’d like for a recessed LED to cast through the button when powered on. This will be for a curved, LED desk lamp made of several glued layers of thin wood veneer. I need to find a solution that allows me to use a thin strip-like sensor that won’t take up more than a few millimetres of depth, all tolled.

I’ve scoured various sections of SparkFun and I’ve come to the conclusion that I need some direction/help.